http://www.ck12.org Chapter 5. Applications of Definite Integrals
A=
∫b
a[f(x)−g(x)]dx
=∫ 3
− 2 [(x+^6 )−(x(^2) )]dx.
Integrating,
A=
[x 2
2 +^6 x−x^3
3] 3
− 2
=^1256.So the area between the two curvesf(x) =x+6 andg(x) =x^2 is 125/ 6.
Sometimes it is possible to apply the area formula with respect to they−coordinates instead of thex−coordinates.
In this case, the equations of the boundaries will be written in such a way thatyis expressed explicitly as a function
ofx(Figure 3).
Figure 3
The Area Between Two Curves(With respect to the y−axis)
Ifwandvare two continuous functions on the interval[c,d]andw(y)≥v(y)for all values ofyin the interval, then
the area of the region that is bounded byx=v(y)on the left,x=w(y)on the right, below byy=c,and above by
y=d,is given by
A=
∫d
c [w(y)−v(y)]dy.Example 2: